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Linear Models for Supervised Learning
Weinan ZhangShanghai Jiao Tong University
http://wnzhang.net
2019 CS420 Machine Learning, Lecture 2
http://wnzhang.net/teaching/cs420/index.html
Discriminative Model and Generative Model
• Discriminative model• modeling the dependence of unobserved variables on
observed ones• also called conditional models.• Deterministic: • Probabilistic:
• Generative model• modeling the joint probabilistic distribution of data• given some hidden parameters or variables
• then do the conditional inference
Discriminative Model and Generative Model
• Discriminative model• modeling the dependence of unobserved variables on
observed ones• also called conditional models.• Deterministic: • Probabilistic:
• Directly model the dependence for label prediction• Easy to define dependence specific features and models• Practically yielding higher prediction performance
• Linear regression, logistic regression, k nearest neighbor, SVMs, (multi-layer) perceptrons, decision trees, random forest etc.
Discriminative Model and Generative Model
• Generative model• modeling the joint probabilistic distribution of data• given some hidden parameters or variables
• then do the conditional inference
• Recover the data distribution [essence of data science]• Benefit from hidden variables modeling
• Naive Bayes, Hidden Markov Model, Mixture Gaussian, Markov Random Fields, Latent Dirichlet Allocation etc.
Linear Regression
Linear Discriminative Models• Discriminative model
• modeling the dependence of unobserved variables on observed ones
• also called conditional models.• Deterministic: • Probabilistic:
• Focus of this course• Linear regression model• Linear classification model
Linear Discriminative Models• Discriminative model
• modeling the dependence of unobserved variables on observed ones
• also called conditional models.• Deterministic: • Probabilistic:
• Linear regression model
y = fμ(x) = μ0 +dX
j=1
μjxj = μ>xy = fμ(x) = μ0 +dX
j=1
μjxj = μ>x
x = (1; x1; x2; : : : ; xd)x = (1; x1; x2; : : : ; xd)
Linear Regression• One-dimensional linear & quadratic regression
Linear Regression Quadratic Regression(A kind of generalizedlinear model)
Linear Regression• Two-dimensional linear regression
Learning Objective• Make the prediction close to the corresponding
label
• Loss function measures the error between the label and prediction
• The definition of loss function depends on the data and task
• Most popular loss function: squared loss
Squared Loss
• Penalty much more on larger distances
• Accept small distance (error) • Observation
noise etc.• Generalization
Least Square Linear Regression
• Objective function to minimize
Minimize the Objective Function• Let N=1 for a simple case, for (x,y)=(2,1)
J(μ) =1
2(y ¡ μ0 ¡ μ1x)2 =
1
2(1¡ μ0 ¡ 2μ1)
2J(μ) =1
2(y ¡ μ0 ¡ μ1x)2 =
1
2(1¡ μ0 ¡ 2μ1)
2
Gradient Learning Methods
μnew à μold ¡ ´@L(μ)
@μμnew à μold ¡ ´
@L(μ)
@μ
Batch Gradient Descent
• Update for the whole batch
Learning Linear Model - Curve
Learning Linear Model - Weights
Stochastic Gradient Descent
• Update for every single instance
J (i)(μ) =1
2(yi ¡ fμ(xi))
2 minμ
1
N
Xi
J (i)(μ)J (i)(μ) =1
2(yi ¡ fμ(xi))
2 minμ
1
N
Xi
J (i)(μ)
μnew = μold ¡ ´ @J(i)(μ)@μμnew = μold ¡ ´ @J(i)(μ)@μ
@J (i)(μ)
@μ= ¡(yi ¡ fμ(xi))
@fμ(xi)
@μ= ¡(yi ¡ fμ(xi))xi
μnew = μold + ´(yi ¡ fμ(xi))xi
@J (i)(μ)
@μ= ¡(yi ¡ fμ(xi))
@fμ(xi)
@μ= ¡(yi ¡ fμ(xi))xi
μnew = μold + ´(yi ¡ fμ(xi))xi
• Compare with BGD• Faster learning• Uncertainty or fluctuation in learning
Linear Classification Model
Mini-Batch Gradient Descent• A combination of batch GD and stochastic GD
• Split the whole dataset into K mini-batches
• Update for each mini-batch
J (k)(μ) =1
2Nk
NkXi=1
(yi ¡ fμ(xi))2J (k)(μ) =
1
2Nk
NkXi=1
(yi ¡ fμ(xi))2
μnew = μold ¡ ´@J (k)(μ)
@μμnew = μold ¡ ´
@J (k)(μ)
@μ
• For each mini-batch k, perform one-step BGD towards minimizing
Mini-Batch Gradient Descent• Good learning stability (BGD)• Good convergence rate (SGD)
• Easy to be parallelized• Parallelization within a mini-batch
Mini-batch
Worker 1
Worker 2
Worker 3
ParallelizedMap Gradient Reduce Gradient Sum
Basic Search Procedure• Choose an initial value for• Update iteratively with the data• Until we research a minimum
μμ
μμ
Basic Search Procedure• Choose a new initial value for• Update iteratively with the data• Until we research a minimum
μμ
μμ
Unique Minimum for Convex Objective
• Different initial parameters and different learning algorithm lead to the same optimum
Convex Set• A convex set S is a set of points such that, given any
two points A, B in that set, the line AB joining them lies entirely within S.
A
BA
B
tx1 + (1¡ t)x2 2 Stx1 + (1¡ t)x2 2 S
for all x1; x2 2 S; 0 · t · 1x1; x2 2 S; 0 · t · 1
Convex set Non-convex set
[Boyd, Stephen, and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.]
Convex Function
is convex if is a convex set andf : Rn ! Rf : Rn ! Rf(tx1 + (1¡ t)x2) · tf(x1) + (1¡ t)f(x2)f(tx1 + (1¡ t)x2) · tf(x1) + (1¡ t)f(x2)
for all
dom fdom f
x1; x2 2 dom f; 0 · t · 1x1; x2 2 dom f; 0 · t · 1
Choosing Learning Rate
• To see if gradient descent is working, print out for each or every several iterations. If does not drop properly, adjust
μnew = μold ¡ ´@J(μ)
@μμnew = μold ¡ ´
@J(μ)
@μ
too smallslow convergence
too largeIncreasing value of
• May overshoot the minimum• May fail to converge• May even diverge
Slide credit Eric Eaton
• The initial point may be too far away from the optimal solution, which takes much time to converge
Algebra Perspective
• Prediction
X =
26664x(1)
x(2)
...
x(n)
37775 =
266664x
(1)1 x
(1)2 x
(1)3 : : : x
(1)d
x(2)1 x
(2)2 x
(2)3 : : : x
(2)d
......
.... . .
...
x(n)1 x
(n)2 x
(n)3 : : : x
(n)d
377775X =
26664x(1)
x(2)
...
x(n)
37775 =
266664x
(1)1 x
(1)2 x
(1)3 : : : x
(1)d
x(2)1 x
(2)2 x
(2)3 : : : x
(2)d
......
.... . .
...
x(n)1 x
(n)2 x
(n)3 : : : x
(n)d
377775 μ =
26664μ1μ2...μd
37775μ =
26664μ1μ2...μd
37775 y =
26664y1
y2...
yn
37775y =
26664y1
y2...
yn
37775
y = Xμ =
26664x(1)μ
x(2)μ...
x(n)μ
37775y = Xμ =
26664x(1)μ
x(2)μ...
x(n)μ
37775• Objective J(μ) =
1
2(y ¡ y)>(y ¡ y) =
1
2(y ¡Xμ)>(y ¡Xμ)J(μ) =
1
2(y ¡ y)>(y ¡ y) =
1
2(y ¡Xμ)>(y ¡Xμ)
Matrix Form
• Gradient
J(μ) =1
2(y ¡Xμ)>(y ¡Xμ) min J(μ)J(μ) =
1
2(y ¡Xμ)>(y ¡Xμ) min J(μ)
@J(μ)
@μ= ¡X>(y ¡Xμ)
@J(μ)
@μ= ¡X>(y ¡Xμ)
@J(μ)
@μ= 0 ) X>(y ¡Xμ) = 0
) X>y = X>Xμ
) μ = (X>X)¡1X>y
@J(μ)
@μ= 0 ) X>(y ¡Xμ) = 0
) X>y = X>Xμ
) μ = (X>X)¡1X>y
• Objective
• Solution
http://dsp.ucsd.edu/~kreutz/PEI-05%20Support%20Files/ECE275A_Viewgraphs_5.pdf
First column
Second column
Matrix Form• Then the predicted values are
y = X(X>X)¡1X>y
= Hy
y = X(X>X)¡1X>y
= Hy
• Geometrical Explanation• The column vectors form a subspace of • H is a least square projection
H: hat matrix
X =
266664x
(1)1 x
(1)2 x
(1)3 : : : x
(1)d
x(2)1 x
(2)2 x
(2)3 : : : x
(2)d
......
.... . .
...
x(n)1 x
(n)2 x
(n)3 : : : x
(n)d
377775 = [x1;x2; : : : ;xd]X =
266664x
(1)1 x
(1)2 x
(1)3 : : : x
(1)d
x(2)1 x
(2)2 x
(2)3 : : : x
(2)d
......
.... . .
...
x(n)1 x
(n)2 x
(n)3 : : : x
(n)d
377775 = [x1;x2; : : : ;xd] y =
26664y1
y2...
yn
37775y =
26664y1
y2...
yn
37775
[x1;x2; : : : ;xd][x1;x2; : : : ;xd] RnRn
ky ¡Xμk2ky ¡Xμk2
More details refer to Sec 3.2. Hastie et al. The elements of statistical learning.
Might be Singular• When some column vectors are not independent
• For example,then is singular, thuscannot be directly calculated.
• Solution: regularization
X>XX>X
x2 = 3x1x2 = 3x1
X>XX>X μ = (X>X)¡1X>yμ = (X>X)¡1X>y
J(μ) =1
2(y ¡Xμ)>(y ¡Xμ) +
¸
2kμk2
2J(μ) =1
2(y ¡Xμ)>(y ¡Xμ) +
¸
2kμk2
2
Matrix Form with Regularization
• Gradient
J(μ) =1
2(y ¡Xμ)>(y ¡Xμ) +
¸
2kμk22 minJ(μ)J(μ) =
1
2(y ¡Xμ)>(y ¡Xμ) +
¸
2kμk22 minJ(μ)
@J(μ)
@μ= ¡X>(y ¡Xμ) + ¸μ
@J(μ)
@μ= ¡X>(y ¡Xμ) + ¸μ
@J(μ)
@μ= 0 ! ¡X>(y ¡Xμ) + ¸μ = 0
! X>y = (X>X + ¸I)μ
! μ = (X>X + ¸I)¡1X>y
@J(μ)
@μ= 0 ! ¡X>(y ¡Xμ) + ¸μ = 0
! X>y = (X>X + ¸I)μ
! μ = (X>X + ¸I)¡1X>y
• Objective
• Solution
Linear Discriminative Models• Discriminative model
• modeling the dependence of unobserved variables on observed ones
• also called conditional models.• Deterministic: • Probabilistic:
• Linear regression with Gaussian noise model
Objective: Likelihood
• Data likelihood
p(²) =1p
2¼¾2e¡
²2
2¾2p(²) =1p
2¼¾2e¡
²2
2¾2
² » N (0; ¾2)² » N (0; ¾2)
p(yjx) =1p
2¼¾2e¡
(y¡μ>x)2
2¾2p(yjx) =1p
2¼¾2e¡
(y¡μ>x)2
2¾2
Learning• Maximize the data likelihood
maxμ
NYi=1
1p2¼¾2
e¡(yi¡μ>xi)
2
2¾2maxμ
NYi=1
1p2¼¾2
e¡(yi¡μ>xi)
2
2¾2
• Maximize the data log-likelihood
logNY
i=1
1p2¼¾2
e¡(yi¡μ>xi)
2
2¾2 =NX
i=1
log1p
2¼¾2e¡
(yi¡μ>xi)2
2¾2
= ¡NX
i=1
(yi ¡ μ>xi)2
2¾2+ const
logNY
i=1
1p2¼¾2
e¡(yi¡μ>xi)
2
2¾2 =NX
i=1
log1p
2¼¾2e¡
(yi¡μ>xi)2
2¾2
= ¡NX
i=1
(yi ¡ μ>xi)2
2¾2+ const
minμ
NXi=1
(yi ¡ μ>xi)2min
μ
NXi=1
(yi ¡ μ>xi)2 Equivalent to least square error learning
Linear Classification
Classification Problem• Given:
• A description of an instance, , where is the instance space.
• A fixed set of categories:
• Determine:• The category of : , where is a
categorization function whose domain is and whose range is
• If the category set binary, i.e. ({false, true}, {negative, positive}) then it is called binary classification.
x 2 Xx 2 X XX
C = fc1; c2; : : : ; cmgC = fc1; c2; : : : ; cmg
xx f(x) 2 Cf(x) 2 C f(x)f(x)XX
CC
C = f0; 1gC = f0; 1g
Binary Classification
Linearly inseparable Non-linearly inseparable
Linear Discriminative Models• Discriminative model
• modeling the dependence of unobserved variables on observed ones
• also called conditional models.• Deterministic:
• Non-differentiable• Probabilistic:
• Differentiable
• For binary classification
y = fμ(x)y = fμ(x)
pμ(yjx)pμ(yjx)
pμ(y = 1jx)pμ(y = 1jx)
pμ(y = 0jx) = 1¡ pμ(y = 1jx)pμ(y = 0jx) = 1¡ pμ(y = 1jx)
Loss Function• Cross entropy loss
0 1 0 0 0
0.1 0.6 0.05 0.05 0.2
Ground Truth
Prediction
L(y; x; pμ) = ¡X
k
±(y = ck) log pμ(y = ckjx)L(y; x; pμ) = ¡X
k
±(y = ck) log pμ(y = ckjx)
H(p; q) = ¡X
x
p(x) log q(x)H(p; q) = ¡X
x
p(x) log q(x)
H(p; q) = ¡Z
xp(x) log q(x)dxH(p; q) = ¡
Zxp(x) log q(x)dx
• For classification problem
±(z) =
(1; z is true
0; otherwise±(z) =
(1; z is true
0; otherwise
Discrete case:
Continuous case:
Cross Entropy for Binary Classification
• Loss function
0 1
0.3 0.7
Ground Truth
Prediction
Class 1 Class 2
L(y; x; pμ) = ¡±(y = 1) log pμ(y = 1jx)¡ ±(y = 0) log pμ(y = 0jx)
= ¡y log pμ(y = 1jx)¡ (1¡ y) log(1¡ pμ(y = 1jx))
L(y; x; pμ) = ¡±(y = 1) log pμ(y = 1jx)¡ ±(y = 0) log pμ(y = 0jx)
= ¡y log pμ(y = 1jx)¡ (1¡ y) log(1¡ pμ(y = 1jx))
Logistic Regression• Logistic regression is a binary classification model
pμ(y = 1jx) = ¾(μ>x) =1
1 + e¡μ>xpμ(y = 1jx) = ¾(μ>x) =
1
1 + e¡μ>x
pμ(y = 0jx) =e¡μ>x
1 + e¡μ>xpμ(y = 0jx) =
e¡μ>x
1 + e¡μ>x
L(y; x; pμ) = ¡y log ¾(μ>x)¡ (1 ¡ y) log(1¡ ¾(μ>x))L(y; x; pμ) = ¡y log ¾(μ>x)¡ (1 ¡ y) log(1¡ ¾(μ>x))
@¾(z)
@z= ¾(z)(1¡ ¾(z))
@¾(z)
@z= ¾(z)(1¡ ¾(z))
@L(y; x; pμ)
@μ= ¡y
1
¾(μ>x)¾(z)(1¡ ¾(z))x¡ (1¡ y)
¡1
1¡ ¾(μ>x)¾(z)(1¡ ¾(z))x
= (¾(μ>x)¡ y)x
μ Ã μ + ´(y ¡ ¾(μ>x))x
@L(y; x; pμ)
@μ= ¡y
1
¾(μ>x)¾(z)(1¡ ¾(z))x¡ (1¡ y)
¡1
1¡ ¾(μ>x)¾(z)(1¡ ¾(z))x
= (¾(μ>x)¡ y)x
μ Ã μ + ´(y ¡ ¾(μ>x))x
• Cross entropy loss function
• Gradient
¾(x)¾(x)
xx
Label Decision• Logistic regression provides the probability
• The final label of an instance is decided by setting a threshold
pμ(y = 1jx) = ¾(μ>x) =1
1 + e¡μ>xpμ(y = 1jx) = ¾(μ>x) =
1
1 + e¡μ>x
pμ(y = 0jx) =e¡μ>x
1 + e¡μ>xpμ(y = 0jx) =
e¡μ>x
1 + e¡μ>x
hh
y =
(1; pμ(y = 1jx) > h
0; otherwisey =
(1; pμ(y = 1jx) > h
0; otherwise
Evaluation Measures
• True / False• True: prediction = label• False: prediction ≠ label
• Positive / Negative• Positive: predict y = 1• Negative: predict y = 0
1 0
1 TruePositive
False Negative
0 False Positive
True Negative
Label
Prediction
Class 1
Class 0
TP: if predicting 1FN: if predicting 0
FP: if predicting 1TN: if predicting 0
Evaluation Measures
• Accuracy: the ratio of cases when prediction = label
1 0
1 TruePositive
False Negative
0 False Positive
True Negative
Label
Prediction
Acc =TP + TN
TP + TN + FP + FNAcc =
TP + TN
TP + TN + FP + FN
Evaluation Measures
• Precision: the ratio of true class 1 cases in those with prediction 1
1 0
1 TruePositive
False Negative
0 False Positive
True Negative
Label
Prediction
Prec =TP
TP + FPPrec =
TP
TP + FP
• Recall: the ratio of cases with prediction 1 in all true class 1 cases
1 0
1 TruePositive
False Negative
0 False Positive
True Negative
Label
Prediction
Rec =TP
TP + FNRec =
TP
TP + FN
Evaluation Measures• Precision-recall tradeoff
• Higher threshold, higher precision, lower recall• Extreme case: threshold = 0.99
• Lower threshold, lower precision, higher recall• Extreme case: threshold = 0
• F1 Measure
y =
(1; pμ(y = 1jx) > h
0; otherwisey =
(1; pμ(y = 1jx) > h
0; otherwise
F1 =2£ Prec£Recall
Prec + RecF1 =
2£ Prec£Recall
Prec + Rec
Evaluation Measures• Ranking-based measure: Area Under ROC Curve (AUC)
Evaluation Measures• Ranking-based measure: Area Under ROC Curve (AUC)
PerfectPrediction
AUC = 1
RandomPredictionAUC = 0.5
Evaluation Measures• A simple example of Area Under ROC Curve (AUC)
False Positive Ratio
TruePositive
Ratio
Prediction Label0.91 10.85 00.77 10.72 10.61 00.48 10.42 00.33 00.25 0.5 0.75 1.0
0.5
1.0
0.75
0.25
AUC = 0.75
Multi-Class Classification
• Still cross entropy loss 0 1 0
0.1 0.7 0.2
Ground Truth
Prediction
L(y; x; pμ) = ¡X
k
±(y = ck) log pμ(y = ckjx)L(y; x; pμ) = ¡X
k
±(y = ck) log pμ(y = ckjx) ±(z) =
(1; z is true
0; otherwise±(z) =
(1; z is true
0; otherwise
• Softmax• Parameters• Can be normalized with m-1 groups of parameters
Multi-Class Logistic Regression• Class set C = fc1; c2; : : : ; cmgC = fc1; c2; : : : ; cmg
pμ(y = cjjx) =eμ>j xPm
k=1 eμ>k x
for j = 1; : : : ;mpμ(y = cjjx) =eμ>j xPm
k=1 eμ>k x
for j = 1; : : : ;m
pμ(y = cjjx)pμ(y = cjjx)• Predicting the probability of
μ = fμ1; μ2; : : : ; μmgμ = fμ1; μ2; : : : ; μmg
Multi-Class Logistic Regression• Learning on one instance
• Maximize log-likelihood
@ log pμ(y = cjjx)
@μj=
@
@μjlog
eμ>j xPm
k=1 eμ>k x
= x¡ @
@μjlog
mXk=1
eμ>k x
= x¡ eμ>j xxPm
k=1 eμ>k x
@ log pμ(y = cjjx)
@μj=
@
@μjlog
eμ>j xPm
k=1 eμ>k x
= x¡ @
@μjlog
mXk=1
eμ>k x
= x¡ eμ>j xxPm
k=1 eμ>k x
maxμ
log pμ(y = cjjx)maxμ
log pμ(y = cjjx)
(x; y = cj)(x; y = cj)
• Gradient
Application Case StudyClick-Through Rate (CTR) Estimation in Online Advertising
[http://news.ifeng.com]
Click or not?
Ad Click-Through Rate Estimation
User Response Estimation Problem
• Problem definition
• Date: 20160320• Hour: 14• Weekday: 7• IP: 119.163.222.*• Region: England• City: London• Country: UK• Ad Exchange: Google• Domain: yahoo.co.uk• URL: http://www.yahoo.co.uk/abc/xyz.html• OS: Windows• Browser: Chrome• Ad size: 300*250• Ad ID: a1890• User occupation: Student• User tags: Sports, Electronics
Click (1) or not (0)?
Predicted CTR (0.15)
One instance data Corresponding label
One-Hot Binary Encoding
• High dimensional sparse binary feature vector• Usually higher than 1M dimensions, even 1B dimensions• Extremely sparse
x=[Weekday=Friday, Gender=Male, City=Shanghai]
x=[0,0,0,0,1,0,0 0,1 0,0,1,0…0]
• A standard feature engineering paradigm
Sparse representation: x=[5:1 9:1 12:1]
Training/Validation/Test Data• Examples (in LibSVM format)
1 5:1 9:1 12:1 45:1 154:1 509:1 4089:1 45314:1 988576:10 2:1 7:1 18:1 34:1 176:1 510:1 3879:1 71310:1 818034:1
…
• Training/Validation/Test data split• Sort data by time• Train:validation:test = 8:1:1• Shuffle training data
Training Logistic Regression• Logistic regression is a binary classification model
pμ(y = 1jx) = ¾(μ>x) =1
1 + e¡μ>xpμ(y = 1jx) = ¾(μ>x) =
1
1 + e¡μ>x
L(y; x; pμ) = ¡y log ¾(μ>x)¡ (1¡ y) log(1¡ ¾(μ>x)) +¸
2jjμjj22L(y; x; pμ) = ¡y log ¾(μ>x)¡ (1¡ y) log(1¡ ¾(μ>x)) +
¸
2jjμjj22
μ Ã (1 ¡ ¸´)μ + ´(y ¡ ¾(μ>x))xμ Ã (1 ¡ ¸´)μ + ´(y ¡ ¾(μ>x))x
• Cross entropy loss function with L2 regularization
• Parameter learning
• Only update non-zero entries
Experimental Results• Datasets
• Criteo Terabyte Dataset• 13 numerical fields, 26 categorical fields• 7 consecutive days out of 24 days in total (about 300 GB)
during 2014• 79.4M impressions, 1.6M clicks after negative down sampling
• iPinYou Dataset• 65 categorical fields• 10 consecutive days during 2013• 19.5M impressions, 937.7K clicks without negative down
sampling
PerformanceModel Linearity
AUC Log LossCriteo iPinYou Criteo iPinYou
Logistic Regression Linear 71.48% 73.43% 0.1334 5.581e-3
Factorization Machine Bi-linear 72.20% 75.52% 0.1324 5.504e-3
Deep NeuralNetworks
Non-linear 75.66% 76.19% 0.1283 5.443e-3
[Yanru Qu et al. Product-based Neural Networks for User Response Prediction. ICDM 2016.]
• Compared with non-linear models, linear models• Pros: standardized, easily understood and implemented, efficient
and scalable• Cons: modeling limit (feature independent assumption), cannot
explore feature interactions
Generalized Linear Models
Review: Linear Regression
• Prediction
X =
26664x(1)
x(2)
...
x(n)
37775 =
266664x
(1)1 x
(1)2 x
(1)3 : : : x
(1)d
x(2)1 x
(2)2 x
(2)3 : : : x
(2)d
......
.... . .
...
x(n)1 x
(n)2 x
(n)3 : : : x
(n)d
377775X =
26664x(1)
x(2)
...
x(n)
37775 =
266664x
(1)1 x
(1)2 x
(1)3 : : : x
(1)d
x(2)1 x
(2)2 x
(2)3 : : : x
(2)d
......
.... . .
...
x(n)1 x
(n)2 x
(n)3 : : : x
(n)d
377775 μ =
26664μ1μ2...μd
37775μ =
26664μ1μ2...μd
37775 y =
26664y1
y2...
yn
37775y =
26664y1
y2...
yn
37775
y = Xμ =
26664x(1)μ
x(2)μ...
x(n)μ
37775y = Xμ =
26664x(1)μ
x(2)μ...
x(n)μ
37775• Objective J(μ) =
1
2(y ¡ y)>(y ¡ y) =
1
2(y ¡Xμ)>(y ¡Xμ)J(μ) =
1
2(y ¡ y)>(y ¡ y) =
1
2(y ¡Xμ)>(y ¡Xμ)
Review: Matrix Form of Linear Reg.
• Gradient
J(μ) =1
2(y ¡Xμ)>(y ¡Xμ) min J(μ)J(μ) =
1
2(y ¡Xμ)>(y ¡Xμ) min J(μ)
@J(μ)
@μ= ¡X>(y ¡Xμ)
@J(μ)
@μ= ¡X>(y ¡Xμ)
@J(μ)
@μ= 0 ! X>(y ¡Xμ) = 0
! X>y = X>Xμ
! μ = (X>X)¡1X>y
@J(μ)
@μ= 0 ! X>(y ¡Xμ) = 0
! X>y = X>Xμ
! μ = (X>X)¡1X>y
• Objective
• Solution
Generalized Linear Models• Dependence
y = f(μ>Á(x))y = f(μ>Á(x))
• Feature mapping function• Mapped feature matrix
© =
2666666664
Á(x(1))
Á(x(2))...
Á(x(i))...
Á(x(n))
3777777775=
2666666664
Á1(x(1)) Á2(x
(1)) ¢ ¢ ¢ Áh(x(1))
Á1(x(2)) Á2(x
(2)) ¢ ¢ ¢ Áh(x(2))...
.... . .
...
Á1(x(i)) Á2(x
(i)) ¢ ¢ ¢ Áh(x(i))...
.... . .
...
Á1(x(n)) Á2(x
(n)) ¢ ¢ ¢ Áh(x(n))
3777777775© =
2666666664
Á(x(1))
Á(x(2))...
Á(x(i))...
Á(x(n))
3777777775=
2666666664
Á1(x(1)) Á2(x
(1)) ¢ ¢ ¢ Áh(x(1))
Á1(x(2)) Á2(x
(2)) ¢ ¢ ¢ Áh(x(2))...
.... . .
...
Á1(x(i)) Á2(x
(i)) ¢ ¢ ¢ Áh(x(i))...
.... . .
...
Á1(x(n)) Á2(x
(n)) ¢ ¢ ¢ Áh(x(n))
3777777775
Matrix Form of Kernel Linear Regression
• Gradient
J(μ) =1
2(y ¡©μ)>(y ¡©μ) min J(μ)J(μ) =
1
2(y ¡©μ)>(y ¡©μ) min J(μ)
@J(μ)
@μ= ¡©>(y ¡©μ)
@J(μ)
@μ= ¡©>(y ¡©μ)
@J(μ)
@μ= 0 ! ©>(y ¡©μ) = 0
! ©>y = ©>©μ
! μ = (©>©)¡1©>y
@J(μ)
@μ= 0 ! ©>(y ¡©μ) = 0
! ©>y = ©>©μ
! μ = (©>©)¡1©>y
• Objective
• Solution
Matrix Form of Kernel Linear Regression
• The optimal parameters with L2 regularization
μ = (©>© + ¸Ih)¡1©>y
= ©>(©©> + ¸In)¡1y
μ = (©>© + ¸Ih)¡1©>y
= ©>(©©> + ¸In)¡1y
for prediction, we never actually need to access ©©
y = ©μ = ©©>(©©> + ¸In)¡1y
= K(K + ¸In)¡1y
y = ©μ = ©©>(©©> + ¸In)¡1y
= K(K + ¸In)¡1y
• With the Algebra trick(P¡1 + B>R¡1B)¡1B>R¡1 = PB>(BPB> + R)¡1(P¡1 + B>R¡1B)¡1B>R¡1 = PB>(BPB> + R)¡1
[http://www.ics.uci.edu/~welling/classnotes/papers_class/Kernel-Ridge.pdf]
where the kernel matrix K = fK(x(i); x(j))gK = fK(x(i); x(j))g
